Whats the difference between Proof, Brilliant Uncirculated and Bullion coins?

The Royal Mint produces three types of uncirculated commemorative coin finish: Proof, Brilliant Uncirculated and Bullion.
Proof coins are the highest standard of commemorative coin produced by The Royal Mint.
Brilliant Uncirculated coins have a higher standard of finish than Bullion coins, without the extra finishing and detail provided on Proof coins.
Bullion coins have a similar standard of finish to circulating coins.